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清华大学：《生物核磁共振波谱学》课程教学资源（文献资料，英文版）The use of nmr methods for conformational

MACINETIC RESONANCE ELSEVIER Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)287-387 The use of nmr methods for conformational studies of nucleic acids Sybren S. Wijmenga*, Bernd N M. van Buuren Umed University, Department of Medical Biochemistry and Biophysics, S 90187 Umed, Sweden

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S.S. Wijmenga, B N.M. van Buuren/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)287-387 299 number of distance constraints affect the precision of fast rotation of the methyl groups and fast internal the determined structure? motion (Eq. (5))are generally incorporated into The intensities of the cross peaks in a NOESY relaxation matrix programs (see below ). More spectrum are related to distances between spins via difficult is the situation for interconversion the relaxation matrix. R conformers, which have distinctly different rotondistances. An example is the ra NOE=Sexp( RTm)=S(I-RTm+ 5(RTm)--.) conversion between N-type and S-type sugars observed in nucleic acids, which leads to very different intra-nucleotide sugar-to-base distances. In Here Tm is the mixing time and s is a scaling factor this case the average distance derived from NOes taken to be equal to 1. The elements Ri of the tends to be heavily biased towards the shorter dis relaxation matrix are given by(see also Section 8) tances. To account for the latter effects requires ensemble averaging, a method which has not beer ky=620)-0 (2) implemented in relaxation matrix programs mainly because of the major computational effort involved where q=r(h/2T)/4, and ry is the distance between Finally, nucleic acids are not spherically but asym protons i and j. For a rigid isotropically tumbling metrically shaped, leading to anisotropic instead of molecule, the spectral density function, (o), can be isotropic tumbling. Consequently, the spectral density written as: function has to be replaced by a more complicated form(see Section 8). Anisotropic tumbling is not con- J(a)= 3) sidered in most relaxation matrix programs(see so that in the slow tumbling limit(at 1)one For short mixing times only the first term in the obtains expansion given in Eq.(1)is required. The isolated spin pair approximation, ISPA, then applies, and the Rn (4) NOE between protons i and j is proportional to their distance to the inverse sixth power: calculations(see below). Internal dynamics or confor- NOE=C mational flexibility implies that interconversion between different conformers takes place, with each Plotting the NOE-I6 versus known distances conformer having a different set of distances. The according to NoE intensity is then derived from the average relaxation rate, (Ri), which is either proportional to NOEi=Cr (1/r), when the motion is slower than the overall tumbling, or to(1/r"), when the averaging is faster allows the determination of the constant C.Distances then the overall tumbling time [ 62]. when the internal can then be calculated via motion is of limited scope (libration motions) the average distance may be approximated by the middle distance. As a result, fast internal libration motions on For longer mixing times spin diffusion becomes more a ps to ns time-scale can be accounted for by using the effective. As a result NOE intensity is lost in cross Lipari and Szabo approach [65, 66] by introducing a peaks involving spins that are close in space, while scaling factor(S)in Eq (4)(see Section 8) NOE intensity may be gained in cross peaks involving Rn=-900=-S spins that are relatively far apart. The net effect is tha versus distance curve flattens and can be described by The fast rotation of the methyl protons also leads to a scaling down of the NOE intensity. Corrections for NOE-6=A+B

number of distance constraints affect the precision of the determined structure? The intensities of the cross peaks in a NOESY spectrum are related to distances between spins via the relaxation matrix, R: NOE ¼ S exp( ¹ Rtm) ¼ S(1 ¹ Rtm þ 1 2 (Rtm) 2 ¹ …) (1) Here tm is the mixing time and S is a scaling factor taken to be equal to 1. The elements Rij of the relaxation matrix are given by (see also Section 8): Rij ¼ q r6 ij (6J(2q) ¹ J(0)) (2) where q ¼ g4 (h/2p) 2 /4, and rij is the distance between protons i and j. For a rigid isotropically tumbling molecule, the spectral density function, J(q), can be written as: J(q) ¼ 2 5 tc 1 þ q2t2 c (3) so that in the slow tumbling limit (qt q 1) one obtains: Rij ¼ ¹ q r6 ij J(0) ¼ ¹ q r6 ij tc (4) This is the equation used in most relaxation matrix calculations (see below). Internal dynamics or conformational flexibility implies that interconversion between different conformers takes place, with each conformer having a different set of distances. The NOE intensity is then derived from the average relaxation rate, hRiji, which is either proportional to h1/r6 i, when the motion is slower than the overall tumbling, or to h1/r3 i 2 , when the averaging is faster then the overall tumbling time [62]. When the internal motion is of limited scope (libration motions) the average distance may be approximated by the middle distance. As a result, fast internal libration motions on a ps to ns time-scale can be accounted for by using the Lipari and Szabo approach [65,66] by introducing a scaling factor (S2 ) in Eq. (4) (see Section 8): Rij ¼ ¹ q r6 ij J(0) ¼ ¹ q r6 ij S2 tc (5) The fast rotation of the methyl protons also leads to a scaling down of the NOE intensity. Corrections for fast rotation of the methyl groups and fast internal motion (Eq. (5)) are generally incorporated into relaxation matrix programs (see below). More difficult is the situation for interconversion between conformers, which have distinctly different proton– proton distances. An example is the rapid interconversion between N-type and S-type sugars observed in nucleic acids, which leads to very different intra-nucleotide sugar-to-base distances. In this case the average distance derived from NOEs tends to be heavily biased towards the shorter distances. To account for the latter effects requires ensemble averaging, a method which has not been implemented in relaxation matrix programs mainly because of the major computational effort involved. Finally, nucleic acids are not spherically but asymmetrically shaped, leading to anisotropic instead of isotropic tumbling. Consequently, the spectral density function has to be replaced by a more complicated form (see Section 8). Anisotropic tumbling is not considered in most relaxation matrix programs (see below). For short mixing times only the first term in the expansion given in Eq. (1) is required. The isolated spin pair approximation, ISPA, then applies, and the NOE between protons i and j is proportional to their distance to the inverse sixth power: NOEij ¼ C 1 r6 ij (6) Plotting the NOE¹1/6 versus known distances, according to NOE ¹ 1=6 ij ¼ C ¹ 1=6 rij (7) allows the determination of the constant C. Distances can then be calculated via rij ¼ C ¹ 1=6 NOE ¹ 1=6 ij (8) For longer mixing times spin diffusion becomes more effective. As a result NOE intensity is lost in cross peaks involving spins that are close in space, while NOE intensity may be gained in cross peaks involving spins that are relatively far apart. The net effect is that the NOE¹1/6 versus distance curve flattens and can be described by NOE ¹ 1=6 ij ¼ A þ B ¹ 1=6 rij (9) S.S. Wijmenga, B.N.M. van Buuren/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 287–387 299

S.S. Wijmenga, B.N.M. van Buuren/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)287-387 The constants a and b can be determined as before used to calculate a new model noe matrix The whole from known distances. This modified ISPa procedure of substitution, back-calculation, etc. is pproach allows the estimation of unknown distances repeated until a good fit between experimental and calculated NOEs is obtained. In the no2Di method 7=(NOE-16 (10) [74] ISPA is used to estimate zeroth-order distances Subsequently, a relaxation matrix is built of all spins Ultimately, all NOE intensity will be uniforml for which measured NOEs are available. To complete spread out through the network of proton spins, and, the relaxation matrix the distances for which no consequently, becomes independent of the distances, zeroth-order distances are obtained are set to a large and distance estimates can no longer be made. This value. The Noe matrix is then back-calculated and method of accounting for spin diffusion is rather from the NOE,(calc) a new distance estimate crude. In effect the method more or less assumes obtained, ry=TiVNOEU(calc)/NOE; (exp). The new that each spin pair is surrounded by a uniform network distance is used to calculate the NOE matrix again, of other spins, which provide the spin diffusion path- giving a new calculated distance, until NOE i(calc ways. Although this approach does take into account and NOEy(exp) are within a certain range(1%) that short and long distances are differently affected This process is repeated for each distance, starting by spin diffusion, it does not allow for the fact that with short distances and working up to the longer different short distances may be differently affected ones, thus obtaining first-order estimates for each by spin diffusion. a rigorously correct approach is to distance. The process can then be repeated until con- take the full relaxation matrix into account, since ulti- vergence is reached. In the no2Di method, obviously mately only the full expression correctly gives the the spin network consists of the other spins for which NOE matrix. The relaxation matrix, R, and thereby an experimental NOE was observed. If the number of the distances, can be calculated from the Noe matrix, experimental NOEs is too sparse the network may if the complete NOE matrix is known become too sparse, and model distances may be included. While in the ' ISPA approach, uni- (11) form surrounding is assumed for each spin pair, here In this way the complete spin network is considered the surrounding of each spin pair does not need to be and spin diffusion can be fully accounted for. One can model distances. It turns out that the ultimate results now in fact derive from the NOE matrix the confor- do not depend strongly on the choice of model [671 mation of the spin network. In practice, not all noe although the best results are obtained with model intensities are known. Consequently, one needs to distances which are as close as possible to the true somehow build up a complete NOE matrix. Several distances [68] approaches have been developed for this purpose What are the possible sources of error in the ISPA [67-74]. These approaches generally proceed as and relaxation matrix approaches? Most easily follows. In the first step a model is chosen, from assessed are errors in the distances resulting from which a model NOE matrix is calculated. Next, the noise and integration errors in the NOE intensity those theoretical NOEs for which measured values The absolute noise in a spectrum can, with modern are available are replaced by experimental NOES, spectrometers, be quite small and varies from 0.01% the rest remain at the model values. Thus, a hybrid to 0.3% depending on the quality of the spectrum model/experimental NOE matrix is constructed which(assuming that the NOE matrix is scaled to 1).In is complete and from which relaxation rates can be addition, there is a relative error from peak integration calculated and distances extracted. In the IRMa [73] which amounts to 10-15% of the peak volume [681 ind the MORaSS [70] approaches these distances are In our studies, peak amplitudes yield good estimates immediately used in a further cycle of simulated of the peak volumes, when determined from highly annealing refinement to obtain new model distances digitized NOESY spectra. The advantage of using and then a new NOE matrix. In the mardigras amplitudes over integration is that errors due to base approach [67] the updated set of distances is directly line distortion and peak overlap are minimized. We

The constants A and B can be determined as before from known distances. This ‘modified’ ISPA approach allows the estimation of unknown distances via rij ¼ (NOE ¹ 1=6 ij ¹ A)B1=6 (10) Ultimately, all NOE intensity will be uniformly spread out through the network of proton spins, and, consequently, becomes independent of the distances, and distance estimates can no longer be made. This method of accounting for spin diffusion is rather crude. In effect the method more or less assumes that each spin pair is surrounded by a uniform network of other spins, which provide the spin diffusion pathways. Although this approach does take into account that short and long distances are differently affected by spin diffusion, it does not allow for the fact that different short distances may be differently affected by spin diffusion. A rigorously correct approach is to take the full relaxation matrix into account, since ultimately only the full expression correctly gives the NOE matrix. The relaxation matrix, R, and thereby the distances, can be calculated from the NOE matrix, if the complete NOE matrix is known: R ¼ ¹ ln(NOE) tm (11) In this way the complete spin network is considered and spin diffusion can be fully accounted for. One can now in fact derive from the NOE matrix the conformation of the spin network. In practice, not all NOE intensities are known. Consequently, one needs to somehow build up a complete NOE matrix. Several approaches have been developed for this purpose [67–74]. These approaches generally proceed as follows. In the first step a model is chosen, from which a model NOE matrix is calculated. Next, those theoretical NOEs for which measured values are available are replaced by experimental NOEs; the rest remain at the model values. Thus, a hybrid model/experimental NOE matrix is constructed which is complete and from which relaxation rates can be calculated and distances extracted. In the IRMA [73] and the MORASS [70] approaches these distances are immediately used in a further cycle of simulated annealing refinement to obtain new model distances and then a new NOE matrix. In the MARDIGRAS approach [67] the updated set of distances is directly used to calculate a new model NOE matrix. The whole procedure of substitution, back-calculation, etc. is repeated until a good fit between experimental and calculated NOEs is obtained. In the NO2DI method [74] ISPA is used to estimate zeroth-order distances. Subsequently, a relaxation matrix is built of all spins for which measured NOEs are available. To complete the relaxation matrix the distances for which no zeroth-order distances are obtained are set to a large value. The NOE matrix is then back-calculated and from the NOEij(calc) a new distance estimate obtained, rij ¼ r 6 ij NOEij(calc)=NOEij(exp) p . The new distance is used to calculate the NOE matrix again, giving a new calculated distance, until NOEij(calc) and NOEij(exp) are within a certain range (1%). This process is repeated for each distance, starting with short distances and working up to the longer ones, thus obtaining first-order estimates for each distance. The process can then be repeated until convergence is reached. In the NO2DI method, obviously the spin network consists of the other spins for which an experimental NOE was observed. If the number of experimental NOEs is too sparse the network may become too sparse, and model distances may be included. While in the ‘modified’ ISPA approach, uniform surrounding is assumed for each spin pair, here the surrounding of each spin pair does not need to be uniform and is built up from both estimated and/or model distances. It turns out that the ultimate results do not depend strongly on the choice of model [67], although the best results are obtained with model distances which are as close as possible to the true distances [68]. What are the possible sources of error in the ISPA and relaxation matrix approaches? Most easily assessed are errors in the distances resulting from the noise and integration errors in the NOE intensity. The absolute noise in a spectrum can, with modern spectrometers, be quite small and varies from 0.01% to 0.3% depending on the quality of the spectrum (assuming that the NOE matrix is scaled to 1). In addition, there is a relative error from peak integration which amounts to 10–15% of the peak volume [68]. In our studies, peak amplitudes yield good estimates of the peak volumes, when determined from highly digitized NOESY spectra. The advantage of using amplitudes over integration is that errors due to base line distortion and peak overlap are minimized. We 300 S.S. Wijmenga, B.N.M. van Buuren/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 287–387

S.S. Wijmenga, B NM van Buuren/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)287-387 found that the peak amplitudes and volumes from a time of 200 ms and a tumbling time of 2 ns. In 3D TOCSY-NOESY spectrum agree within 10%.A addition, 0. 1% random noise was added to the noe second source of possible error is that in practice, intensities and 20% of the NOEs were deleted. Figure NOESY spectra are recorded with relatively short 7(a) of van de ven et al. [74] gives the initial ISPA relaxation delays(RD)inbetween FID recordings, estimates of the distances, while Fig. 7(b) and(c) i.e. RD<5T- This will affect the signal amplitude represent distance estimates after five cycles of from different spins differently, depending on their T1 NO2DI refinement using the IsPa as starting values relaxation rates. Such effects are generally not or a wrong model as source for starting values of the accounted for in the above described programs distances, respectively. van de ven et al. found that third, more important issue, is internal dynami the initial distance estimates. obtained via the IsPa Fast internal libration motions on a ps to ns time- approach, have in fact reasonably small error bounds, scale can be accounted for via Lipari and Szabo's the distances are correct within +0.3 A at 2.0 A up to formalism [65, 66] by introducing a scaling factor +0.6 A at 5 A (as judged from Fig. 7(a)in Ref. [74] ()(see Section 8). The fast rotation of the methyl As expected the shorter distances are being calculated protons also leads to a scaling down of the NOE inten- too large and the longer distances calculated too short sity. These corrections are generally incorporated into on average. A'modified'ISPA estimate of the dis- the programs discussed above. To account for inter tances removes this bias and leads to narrower error conversion between conformers with distinctly bounds of +0.2 A at 2.0 A to +0.6 A at 5A different proton-proton distances requires ensemble (again as judged from Fig. 7(a)in Ref [74]). Most averaging, a method which has not been implemented importantly, we note that all distances are correctl in the programs discussed above, mainly because of estimated within these bounds. Van de ven et al. find the enormous computational effort involved. The that after five rounds of relaxation matrix refinement fourth, and major, source of error in the derivation the shorter distances are now correctly estimated of distances from NOEs seems to originate from the (within + 0.2 A for distances up to 3.0 A),while fact that the more spin diffusion contributes to the for larger distances the majority of the estimated dis- intensity of a cross peak, the more difficult it becomes tances have errors up to +0.7 A at 5 A, which is the to retrieve the direct contribution, even when relaxa- same order of magnitude as the error when using the tion matrix approaches are used. This can be under- ISPA estimates Similar error bounds for ISPA and stood qualitatively from the decreased dependence on relaxation matrix refined distances have been obtained distance, which is found when spin diffusion plays a by Borgias and James using MARDIGRAS [67] role. This has a profound effect on how errors in noe NOE data with 0.3% error. Thus, it seems that rather ntensities translate into errors in the derived dis- similar error bounds are obtained for 'modified " ISPa tances. At short mixing times, when ISPa applies derived and relaxation matrix refined distances at errors in the noe intensity lead to strongly damped least under these conditions. In view of these results errors in the derived distances. because of the inverse we have investigated the derivation of distances from sixth power relationship. On the other hand, when NOE data for the case of a 3D TOCSY-NOESY spec- spin diffusion becomes highly effective, the distance trum of a 12-mer RNA duplex [75-77]. The NOE dependence of the NOEs becomes less strong, and the mixing time was 200 ms, so that spin diffusion does error in the noe translates into much larger errors in affect the Noes, albeit to a limited extent. The NOe the derived distances intensities derived from the 3D TOCSY-NOESY How accurately can distances be obtained and how spectrum were corrected for T2 effects and missing much better does a relaxation matrix approach TOCSY peaks. Distances were then estimated form as compared to a simple ISPA approach? The a 'modified ISPA approach; the NOE- intensities ultimate errors in the distance estimates are nicely for known distances were plotted along the y axis illustrated by the model calculations performed against the known distances along the x axis the van de ven et al. [74] and Borgias and James [671 known distances covered the complete 1.8 to 5.0A In the calculation of van de Ven et al., the NOEs of a range. Calibration was done by fitting a straight line B-DNA duplex were calculated assuming a mixing through the data according to Eq.(9). The known

found that the peak amplitudes and volumes from a 3D TOCSY–NOESY spectrum agree within 10%. A second source of possible error is that in practice, NOESY spectra are recorded with relatively short relaxation delays (RD) inbetween FID recordings, i.e. RD , 5 T1. This will affect the signal amplitude from different spins differently, depending on their T1 relaxation rates. Such effects are generally not accounted for in the above described programs. A third, more important issue, is internal dynamics. Fast internal libration motions on a ps to ns timescale can be accounted for via Lipari and Szabo’s formalism [65,66] by introducing a scaling factor (S2 ) (see Section 8). The fast rotation of the methyl protons also leads to a scaling down of the NOE intensity. These corrections are generally incorporated into the programs discussed above. To account for interconversion between conformers with distinctly different proton–proton distances requires ensemble averaging, a method which has not been implemented in the programs discussed above, mainly because of the enormous computational effort involved. The fourth, and major, source of error in the derivation of distances from NOEs seems to originate from the fact that the more spin diffusion contributes to the intensity of a cross peak, the more difficult it becomes to retrieve the direct contribution, even when relaxation matrix approaches are used. This can be understood qualitatively from the decreased dependence on distance, which is found when spin diffusion plays a role. This has a profound effect on how errors in NOE intensities translate into errors in the derived distances. At short mixing times, when ISPA applies, errors in the NOE intensity lead to strongly damped errors in the derived distances, because of the inverse sixth power relationship. On the other hand, when spin diffusion becomes highly effective, the distance dependence of the NOEs becomes less strong, and the error in the NOE translates into much larger errors in the derived distances. How accurately can distances be obtained and how much better does a relaxation matrix approach perform as compared to a simple ISPA approach? The ultimate errors in the distance estimates are nicely illustrated by the model calculations performed by van de Ven et al. [74] and Borgias and James [67]. In the calculation of van de Ven et al., the NOEs of a B-DNA duplex were calculated assuming a mixing time of 200 ms and a tumbling time of 2 ns. In addition, 0.1% random noise was added to the NOE intensities and 20% of the NOEs were deleted. Figure 7(a) of van de Ven et al. [74] gives the initial ISPA estimates of the distances, while Fig. 7(b) and (c) represent distance estimates after five cycles of NO2DI refinement using the ISPA as starting values or a wrong model as source for starting values of the distances, respectively. Van de Ven et al. found that the initial distance estimates, obtained via the ISPA approach, have in fact reasonably small error bounds; the distances are correct within 6 0.3 A˚ at 2.0 A˚ up to 6 0.6 A˚ at 5 A˚ (as judged from Fig. 7(a) in Ref. [74]. As expected the shorter distances are being calculated too large and the longer distances calculated too short on average. A ‘modified’ ISPA estimate of the distances removes this bias and leads to narrower error bounds of , 6 0.2 A˚ at 2.0 A˚ to 6 0.6 A˚ at 5 A˚ (again as judged from Fig. 7(a) in Ref. [74]). Most importantly, we note that all distances are correctly estimated within these bounds. Van de Ven et al. find that after five rounds of relaxation matrix refinement the shorter distances are now correctly estimated (within 6 0.2 A˚ for distances up to 3.0 A˚ ), while for larger distances the majority of the estimated distances have errors up to 6 0.7 A˚ at 5 A˚ , which is the same order of magnitude as the error when using the ISPA estimates. Similar error bounds for ISPA and relaxation matrix refined distances have been obtained by Borgias and James using MARDIGRAS [67] for NOE data with 0.3% error. Thus, it seems that rather similar error bounds are obtained for ‘modified’ ISPA derived and relaxation matrix refined distances, at least under these conditions. In view of these results we have investigated the derivation of distances from NOE data for the case of a 3D TOCSY–NOESY spectrum of a 12-mer RNA duplex [75–77]. The NOE mixing time was 200 ms, so that spin diffusion does affect the NOEs, albeit to a limited extent. The NOE intensities derived from the 3D TOCSY–NOESY spectrum were corrected for T2 effects and missing TOCSY peaks. Distances were then estimated using a ‘modified’ ISPA approach; the NOE¹1/6 intensities for known distances were plotted along the y axis against the known distances along the x axis; the known distances covered the complete 1.8 to 5.0 A˚ range. Calibration was done by fitting a straight line through the data according to Eq. (9). The known S.S. Wijmenga, B.N.M. van Buuren/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 287–387 301

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